Microwave spin resonance in epitaxial thin films of spin liquid candidate TbInO3

This study employs coplanar superconducting resonators to probe magnetic excitations in epitaxial TbInO3 thin films, revealing extreme magnetic frustration down to 20 mK and elucidating the complex interplay of spin-orbit coupling, crystal fields, and improper ferroelectricity that shapes its quantum spin liquid candidate ground state.

The Gist

Imagine you have a huge crowd of people (atoms) in a room, and you want them to all stand in perfect, orderly rows, facing the same direction. This is what happens in most magnets: the "magnetic moments" (tiny internal compasses) line up neatly.

But in a special class of materials called Quantum Spin Liquids (QSLs), the rules are different. The room is designed so that no matter how hard the people try, they can't all agree on a direction. They are stuck in a state of constant, chaotic disagreement. This is called frustration. Instead of freezing into a solid, orderly pattern, they keep "fluctuating" or dancing around, even when the room is ice cold.

This paper is about a scientist's team trying to catch a glimpse of this chaotic dance in a specific material called TbInO3 (pronounced "T-b-In-O-three").

Here is the story of how they did it, explained simply:

1. The Problem: The Material is Too Small to Measure

Usually, to study these magnetic dances, scientists need a big chunk of the material (a crystal). They use giant machines like neutron scatterers (which are like massive X-ray machines) to see how the atoms move.

But the team had something new: thin films. Imagine a layer of TbInO3 so thin it's like a sheet of paper, or even thinner. It's so small that the giant machines can't see it. It's like trying to hear a whisper in a stadium using a microphone meant for a rock concert.

2. The Solution: The "Super-Sensitive Ear"

To solve this, the team built a microwave resonator. Think of this as a super-tiny, super-sensitive musical instrument (like a guitar string) made of superconducting metal.

• How it works: They placed the thin film right on top of this instrument.
• The Trick: When they sent microwave signals through the instrument, it started to "hum" at a very specific pitch.
• The Interaction: The magnetic atoms in the thin film are like tiny magnets. When the team applied a magnetic field, these atoms tried to flip their orientation. If the "hum" of the instrument matched the energy needed to flip an atom, the atoms would absorb some of the energy.
• The Result: The instrument's "hum" would get slightly quieter (it would decay faster). By listening to how the hum changed, the scientists could hear the atoms flipping, even though there were so few of them.

3. The Discovery: A Tale of Two Groups

When they listened to the thin film, they didn't just hear one sound. They heard two distinct groups of atoms behaving differently.

• The Mystery: Why two groups? It turns out that in TbInO3, the atoms aren't all sitting in identical chairs. Because of a weird electrical quirk in the material (called "improper ferroelectricity"), the atoms are shifted slightly.
• The Analogy: Imagine a dance floor where 2/3 of the dancers are standing on flat tiles, and 1/3 are standing on slightly tilted tiles. Even though they are all doing the same dance, the ones on the tilted tiles have to move their arms differently.
• The Finding: The team found that the atoms on the "flat tiles" and the "tilted tiles" had different "g-factors" (a scientific way of saying they respond to magnetic fields in different ways). They confirmed that the material has two distinct flavors of magnetic atoms, and they are both still dancing (fluctuating) even at temperatures near absolute zero.

4. The Big Picture: Why Does This Matter?

The team measured how "frustrated" these atoms are. They found that the atoms refuse to line up even when the temperature is 220 times lower than the temperature where they should have frozen into order.

• The Frustration Index: Think of this as a score of how stubborn the atoms are. A score of 220 is incredibly high. It suggests that TbInO3 is a prime candidate for being a Quantum Spin Liquid.
• Why we care: Quantum Spin Liquids are weird states of matter that might hold the key to future quantum computers. They are like a "soup" of quantum information that doesn't break down easily. Understanding them helps us build better technology.

Summary

The scientists took a material that is too thin to be studied by normal methods, put it on a super-sensitive microwave "ear," and listened to its magnetic heartbeat. They discovered that the atoms inside are incredibly stubborn, refusing to settle down even at the coldest temperatures, and they come in two distinct "flavors" due to the material's unique structure. This confirms that TbInO3 is a very strong candidate for a Quantum Spin Liquid, a state of matter that could revolutionize how we think about physics and computing.

Technical Summary

1. Problem Statement

The identification of Quantum Spin Liquid (QSL) states in real materials is challenging because these states lack a conventional order parameter (no long-range magnetic order). While theoretical models predict QSLs in frustrated lattices (e.g., triangular, kagome), experimental verification requires detecting the absence of ordering down to temperatures far below the characteristic exchange energy scale ($\theta_{CW}$).

Existing techniques face significant limitations when applied to epitaxial thin films:

• Volume Constraints: Conventional bulk techniques like neutron scattering and specific heat measurements require large sample volumes, which are unavailable in thin films.
• Temperature Limits: Previous magnetic susceptibility measurements on TbInO3 thin films using SQUIDs were limited to temperatures above ~400 mK (in-plane) or showed ordering signatures around 1 K (out-of-plane), failing to probe the deep milliKelvin regime where QSL behavior is most distinct.
• Material Complexity: TbInO3 is a candidate QSL material with unique properties: a frustrated 2D triangular lattice of Tb$^{3+}$ moments, strong spin-orbit coupling, and improper ferroelectricity (where Tb sites are spatially shifted, creating two distinct local environments). Understanding how these factors combine to shape the ground state requires a probe capable of high sensitivity at ultra-low temperatures.

2. Methodology

The authors developed a novel measurement technique adapting circuit quantum electrodynamics (cQED) principles to probe magnetic excitations in thin films.

• Sample Preparation:
  • Epitaxial thin films of TbInO$_3$ were grown on yttria-stabilized zirconia (YSZ) substrates using Molecular Beam Epitaxy (MBE).
  • Superconducting NbTiN films were deposited via magnetron sputtering and patterned into coplanar waveguide (CPW) $\lambda/4$ resonators using photolithography and reactive ion etching.
• Experimental Setup:
  • Microwave Resonance: The device acts as a high-quality factor ($Q \sim 10^5$) harmonic oscillator. An external static magnetic field ($B_0$) is applied in-plane, perpendicular to the microwave magnetic field ($B_{mw}$).
  • Detection Mechanism: When the Zeeman splitting of the spins ($\Delta E = g\mu_B B_0$) matches the resonator frequency ($hf$), resonant energy exchange occurs between photons and spins. This increases the resonator's decay rate ($\kappa$), which is measured via the transmission coefficient ($S_{21}$).
  • Background Subtraction: To isolate the film's signal, a control resonator was fabricated on the same chip where the TbInO$_3$ film was removed via ion milling, allowing for the subtraction of substrate (YSZ) and interface defects.
• Temperature Range: Measurements were conducted in a dilution refrigerator, reaching base temperatures as low as 18 mK.

3. Key Contributions

1. Technique Development: Established a robust method for probing magnetic susceptibility and spin dynamics in insulating thin films with volumes too small for bulk techniques, extending measurements down to the 20 mK regime.
2. Identification of Dual g-Factors: Discovered two distinct, broad spin resonance features in TbInO$_3$ with $g$-factors of 5.1 and 6.6. This was attributed to the improper ferroelectric distortion of the crystal lattice, which splits the Tb sites into two non-equivalent classes (in a 2:1 ratio) with distinct local crystal fields.
3. Ground State Modeling: Developed a crystal-field model showing that the ground state is a non-Kramers doublet formed by superpositions of $m_J$ states ($|m_J| = 1, 2$ or $4, 5$). The model successfully explains the observed $g$-factors based on the mixing angles of these states.
4. Quantification of Frustration: Provided the most stringent test of magnetic ordering in TbInO$_3$ to date, confirming the absence of magnetic order down to 20 mK.

4. Key Results

• Spin Resonance Spectrum:
  • The spectrum revealed three broad features:
    • R1 ($g \approx 2.1$): Identified as originating from defects/impurities in the YSZ substrate.
    • R2 ($g = 5.1 \pm 0.9$) and R3 ($g = 6.6 \pm 0.5$): Originating from the TbInO$_3$ film.
  • The amplitude ratio of R2 to R3 is approximately 2:1, consistent with the structural model where 2/3 of Tb sites occupy one environment and 1/3 occupy another due to ferroelectric distortion.
  • Narrow lines at $g \approx 1.9, 2.3,$ and $4.1$ were identified as defects in the superconductor, interface oxygen vacancies, and Fe impurities in the substrate, respectively.
• Magnetic Susceptibility:
  • By integrating the spectral weight of the resonance, the authors extracted the in-plane magnetic susceptibility ($\chi_{ab}$).
  • The susceptibility follows a Curie-Weiss law down to 50 mK and shows no sign of saturation or ordering down to 20 mK.
  • The low-temperature Curie-Weiss temperature is $\theta_{CW}^{LT} \approx -86$ mK, significantly lower than the high-temperature value ($\sim -11$ K), indicating strong renormalization of interactions at low $T$.
• Frustration Index:
  • The frustration index $f = |\theta_{CW}| / T_{ord}$ was calculated. With no ordering observed down to 20 mK, $f > 220$. This is one of the highest frustration indices reported for QSL candidates, strongly supporting the spin liquid hypothesis.
• Anisotropy:
  • Combining these in-plane results with previous out-of-plane SQUID data (which showed a peak at ~1 K and no exponential drop), the authors conclude Tb moments have easy-plane anisotropy.

5. Significance

• Validation of QSL Candidate: The work provides compelling evidence that TbInO$_3$ is a robust QSL candidate, with magnetic moments fluctuating down to temperatures two orders of magnitude lower than the exchange energy scale, without developing long-range order.
• Role of Ferroelectricity: The study uniquely demonstrates how improper ferroelectricity (structural distortion) directly influences magnetic properties by creating distinct magnetic sublattices with different $g$-factors, a feature specific to TbInO$_3$ among spin liquid candidates.
• Methodological Impact: The technique of using superconducting resonators to probe thin-film frustrated magnets opens new avenues for studying QSLs in heterostructures, interfaces, and strained films, where traditional bulk probes are inapplicable.
• Theoretical Connection: The findings suggest a potential link to the Kitaev honeycomb model due to the anisotropic exchange interactions on the distorted lattice, positioning TbInO$_3$ alongside materials like $\alpha$-RuCl$_3$ as a platform for exploring exotic quantum phases.

In summary, this paper successfully bridges the gap between thin-film growth and ultra-low-temperature magnetic spectroscopy, confirming the exotic, highly frustrated nature of the TbInO$_3$ ground state and establishing a new standard for characterizing quantum spin liquids in reduced dimensions.